We introduce Modular Origami Halftoning, a set of techniques for representing continuous-tone greyscale images using a fixed set of origami modules folded from two-coloured paper. By arranging the modules to reveal different amounts of the paper's coloured side, each module can approximate the brightness of a single pixel from an image. We present two variations based on affixing a grid of disconnected modules to a backing, and one variation in which the modules interlock into a carpet, requiring no tape or glue.